Fractal properties of Aldous-Kendall random metric

TL;DR

This paper studies the fractal geometry of a scale-invariant random metric on Euclidean space generated by a Poisson line process, revealing its Hausdorff dimension and multifractal properties.

Contribution

It proves the Hausdorff dimension of the Aldous-Kendall random metric space exceeds Euclidean dimension and demonstrates its multifractal nature, confirming a prior conjecture.

Findings

Hausdorff dimension is (b3-1)d/(b3-d) > d

The space exhibits multifractal properties with some points having large surrounding balls

Confirms a conjecture of Kahn about the fractal dimension

Abstract

Investigating a model of scale-invariant random spatial network suggested by Aldous, Kendall constructed a random metric $T$ on $\mathbb{R}^d$, for which the distance between points is given by the optimal connection time, when travelling on the road network generated by a Poisson process of lines with a speed limit. In this paper, we look into some fractal properties of that random metric. In particular, although almost surely the metric space $\left(\mathbb{R}^d,T\right)$ is homeomorphic to the usual Euclidean $\mathbb{R}^d$, we prove that its Hausdorff dimension is given by $(\gamma-1)d/(\gamma-d)>d$, where $\gamma>d$ is a parameter of the model; which confirms a conjecture of Kahn. We also find that the metric space $\left(\mathbb{R}^d,T\right)$ equipped with the Lebesgue measure exhibits a multifractal property, as some points have untypically big balls around them.